This is a very good question. As far as I know, any textbook writer writes their own problems or exercises. If the textbook covers an exam then they often use old exam questions as part of the exercises. Exam markers often double their income as question setters. This happens with "A" levels in UK.
I don't know about English language textbooks specifically, but I wrote two textbooks in another field, and can tell you that the exercises formed, by far, the hardest part. It is a work of art to include questions that go gradually from very easy to very hard, so giving the reader more and more confidence and ability solving them; the whole point of the exercise.
I don't know if exercise compilers have a special name.